Algebraic Proof of 4-Colour Theorem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T03:43:48Z http://mathoverflow.net/feeds/question/19240 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19240/algebraic-proof-of-4-colour-theorem Algebraic Proof of 4-Colour Theorem? Tony Huynh 2010-03-24T21:59:25Z 2012-10-20T16:16:19Z <p><strong>4-Colour Theorem.</strong> Every planar graph is 4-colourable. </p> <p>This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism because it heavily relied on the use of computers. The situation was partially remedied 20 years later, when Robertson, Sanders, Seymour, and Thomas published a new proof of the theorem. This new proof still relied on computer analysis, but to such a lower extent that their proof was actually verifiable. Finally, in 2005, Gonthier and Werner used the <a href="http://en.wikipedia.org/wiki/Coq" rel="nofollow">Coq</a> proof assistant to formalize a proof, so I suppose only the most die hard skeptics remain.</p> <p>My question stems from reading this <a href="http://people.math.gatech.edu/~thomas/PAP/update.pdf" rel="nofollow">paper</a> by Robin Thomas. In it, he describes several interesting reformulations of the 4-colour theorem. Here is one:</p> <p>Note that the cross-product on vectors in $\mathbb{R}^3$ is not an associative operation. We therefore define a <em>bracketing</em> of a cross-product $v_1 \times \dots v_n$ to be a set of brackets which makes the product well-defined. </p> <p><strong>Theorem.</strong> Let $i, j, k$ be the standard unit vectors in $\mathbb{R}^3$. For any two different bracketings of the product $v_1 \times \dots \times v_n$, there is an assignment of $i,j,k$ to $v_1, \dots, v_n$ such that the two products are equal and non-zero. </p> <p>The surprising fact is that this innocent looking theorem implies the 4-colour theorem. </p> <p><strong>Question.</strong> Is anyone working on an algebraic proof of the 4-colour theorem (say by trying to prove the above theorem)? If so, what techniques are involved? What partial progress has been made? Or do most people consider the effort/reward ratio of such an endeavor to be too high?</p> <p>I think it would be interesting to have an algebraic proof, even a very long one, particularly if the algebraic proof does not use computers. Given its connection to many other areas (Temperley-Lieb Algebras), the problem seems to be amenable to other forms of attack. </p> http://mathoverflow.net/questions/19240/algebraic-proof-of-4-colour-theorem/19242#19242 Answer by David Lehavi for Algebraic Proof of 4-Colour Theorem? David Lehavi 2010-03-24T22:08:18Z 2010-03-24T22:08:18Z <p>Does <a href="http://front.math.ucdavis.edu/q-alg/9606016" rel="nofollow">"Lie Algebras and the Four Color Theorem"</a> by Dror Bar-Natan qualify ?</p> http://mathoverflow.net/questions/19240/algebraic-proof-of-4-colour-theorem/19274#19274 Answer by Igor Pak for Algebraic Proof of 4-Colour Theorem? Igor Pak 2010-03-25T04:19:58Z 2010-03-25T04:19:58Z <p>There is a classical approach by Birkhoff and Lewis, which remained dormant for decades. It was recently revived by Cautis and Jackson (start <a href="http://www.math.columbia.edu/~scautis/papers/TL.pdf" rel="nofollow">here</a> and proceed <a href="http://www.math.columbia.edu/~scautis/papers/tuttefinal.pdf" rel="nofollow">here</a>), using the Temperley-Lieb algebra. </p> http://mathoverflow.net/questions/19240/algebraic-proof-of-4-colour-theorem/19336#19336 Answer by Noah Snyder for Algebraic Proof of 4-Colour Theorem? Noah Snyder 2010-03-25T17:57:53Z 2010-03-25T17:57:53Z <p>I must admit I'm a bit baffled about what the <em>question</em> is here, and about why so many people have voted it up. What are you looking for in an answer? I don't think it's appropriate to post speculation on the internet about which mathematicians are privately working on which big problems. As to public work, you seem to have a weirdly restrictive view of what "working on" and "partial progress" mean that don't fit with my understanding of how mathematics works. Several papers have been written on the subject of possible algebraic proofs of the 4-color theorem (look at google scholar or Mathscinet for <a href="http://scholar.google.com/scholar?cites=17068357875107999385&amp;hl=en&amp;as_sdt=20000000000" rel="nofollow">papers which cite the Saleur-Kauffman paper</a> mentioned in the paper you're reading), but if the Bar-Natan paper doesn't count for you then you're likely to be disappointed by all of them.</p> <p>The long and short of it is that everyone in quantum topology would love to prove the 4-color theorem and occasionally thinks about it. There's lots of tantalizing clues that an algebraic argument has promise, but if anyone knew how to prove it they'd have done so. As far as I know there isn't anyone who is holed up in their attic thinking about only the 4-color theorem, instead there's a lot of people who every time they find a new tool think "hrm, I wonder if this tool would work on the 4-color theorem?"</p> http://mathoverflow.net/questions/19240/algebraic-proof-of-4-colour-theorem/20194#20194 Answer by guest for Algebraic Proof of 4-Colour Theorem? guest 2010-04-02T22:13:16Z 2010-04-02T22:13:16Z <p>An article in Scientific American, Jan 2003 offered a supposed counterexample, that sparked my interest in the problem. That and the complexity of the Appel and Haken proof motivated me to do my own study. It has minimal math, but is consistent, and approaches the problem from an entirely different direction. If interested it's at insight.awardspace.info.</p> http://mathoverflow.net/questions/19240/algebraic-proof-of-4-colour-theorem/110165#110165 Answer by Anatoly for Algebraic Proof of 4-Colour Theorem? Anatoly 2012-10-20T16:16:19Z 2012-10-20T16:16:19Z <p>Algebraic Proof of 4-Colour Theorem? see please <a href="http://www.math.accent.kiev.ua/article/00/4ct-2-.htm" rel="nofollow">http://www.math.accent.kiev.ua/article/00/4ct-2-.htm</a></p>